Calculate the area of the triangle whose sides are 18 cm, 24 cm and 30 cm in length. Also, find the length of the altitude corresponding to the smallest side.

Answer

To calculate the area of the triangle with sides 18 cm, 24 cm, and 30 cm, we will use Heron's formula. Let's go through the steps:

### Step 1: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of a triangle is given by the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.

For our triangle:
- \( a = 18 \) cm
- \( b = 24 \) cm
- \( c = 30 \) cm

Calculating \( s \):
\[ s = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \text{ cm} \]

### Step 2: Apply Heron's formula to find the area (A)
Heron's formula for the area \( A \) of a triangle is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

Substituting the values:
\[ A = \sqrt{36(36-18)(36-24)(36-30)} \]
Calculating each term:
\[ A = \sqrt{36 \times 18 \times 12 \times 6} \]

### Step 3: Simplify the expression
Calculating inside the square root:
\[ A = \sqrt{36 \times 18 \times 12 \times 6} \]

Calculating step by step:
- \( 36 = 6^2 \)
- \( 18 = 6 \times 3 \)
- \( 12 = 6 \times 2 \)
- \( 6 = 6 \)

Thus:
\[ A = \sqrt{6^2 \times (6 \times 3) \times (6 \times 2) \times 6} \]
\[ = \sqrt{6^5 \times 6} = \sqrt{6^6} = 6^3 = 216 \text{ cm}^2 \]

### Step 4: Find the altitude corresponding to the smallest side
The smallest side is 18 cm. The area can also be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Using the base as 18 cm:
\[ 216 = \frac{1}{2} \times 18 \times h \]
Solving for \( h \):
\[ 216 = 9h \implies h = \frac{216}{9} = 24 \text{ cm} \]

### Final Answers:
- The area of the triangle is \( 216 \text{ cm}^2 \).
- The length of the altitude corresponding to the smallest side (18 cm) is \( 24 \text{ cm} \).